My Chemistry teacher couldn't help me out with this, so I thought I'd ask here.

Since heating a metal makes it more conductive and more able to contain electrons in its metallic bonds, is it possible to heat a rechargeable battery (verrrrry carefully, of course) and charge it at that temperature, then cool it back down and be left with a battery that contains more electrons in its anode than typical (and thus able to discharge more energy), like supersaturating an aqueous solution with ions?

TL;DR - Can you:1- Heat a battery2- Charge it3- Cool it down4- Get more electricity?

Echoing the others, I'm pretty sure it's a definitively bad idea. How bad depends on the type of battery, how much heat you apply and how much you attempt to overcharge the battery. Results may range from battery degradation and failure to explosion and being splashed with acid. Remember, you're not just dealing with the physical effect of heat on metal conductors, you're dealing with the chemical effects of applying heat and current to a set of redox reactions which may be taking place in a system with other components of varying stability. Might end up boiling off electrolyte or solvent and ruining the battery's ability to charge or discharge later! Might end up plating additional metal and short-circuiting the battery! Might generate explosive gases which would definitely ruin your day if not properly vented! All of this also assumes you've overcome whatever technology the battery manufacturer has put in place to prevent you from doing exactly this thing that you intend to do. So don't do it.

- Battery capacity has nothing to do with conductivity;- I have no idea why heating a metal would somehow increase its ability to "contain electrons". I strongly suspect it's the opposite;- Even if it does, in a battery you don't store electrons. You store chemical energy in compounds that can go redox in a way that generates voltage. So in theory, the battery's capacity is determined simply by the amount of chemicals that can go redox available inside it. Nothing to do with temperature, and nothing to do with how many surplus electrons are in the metal parts. (actually, there is some energy in it, but an irrelevant amount. Trying to get usable energy out of surplus charge in metal parts is like trying to power a cellphone with your clothes' static electricity)